Manufacturing method and manufacturing apparatus of pipe, thickness deviation information derivation apparatus, and computer program

ABSTRACT

A complex Fourier transform is performed on measured values of wall thickness at points in a cross section of a pipe in an axial direction, types of thickness deviation are classified, thickness deviation amount is calculated from the absolute value of the complex Fourier component, the position of a thick or thin portion of thickness deviation is calculated from the phase of the complex Fourier component, and manufacturing conditions of the pipe are adjusted based on these variables. The relationship r·exp(jθ) between the thickness deviation amount r and phase θ of a first-order thickness deviation obtained for cross sections in the axial direction is subjected to a complex Fourier transform as a function of pipe longitudinal direction, the thickness deviation is further classified by the frequency of twist of the thickness deviation, and an action is taken to prevent the thickness deviation according to the classified thickness deviation.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of PCT International Application No.PCT/JP2004/003238 which has an International filing date of Mar. 11,2004 and designated the United States of America.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a manufacturing method/manufacturingapparatus for manufacturing a seamless pipe while preventing occurrenceof thickness deviations, a thickness deviation information derivationapparatus for deriving information about a thickness deviation, and acomputer program for realizing a computer as the thickness deviationinformation derivation apparatus.

2. Description of Related Art

A typical example of a manufacturing method of seamless pipes is amethod using a mandrel mill. In this method, first, a billet heated to arequired temperature in a heating furnace is pierced and rolled by apiercing mill to obtain a hollow shell, next the hollow shell is drawnand rolled using the mandrel mill, and then sizing rolling to adjust theexternal diameter and wall thickness is performed using a reducing millor a sizing mill.

In a seamless pipe manufactured by the above-mentioned process, varioustypes of thickness deviations as shown in FIG 1A and FIG. 1B may occur.FIG. 1A and FIG. 1B are cross sectional views of a seamless pipe withvarious types of thickness deviations. FIG. 1A shows cross sections inthe direction of the axis of the seamless pipe, and FIG. 1B shows across section including the central axis of the seamless pipe. Variousfactors are listed as the causes of the thickness deviations. Amongthem, as a factor related to manufacturing equipment, for example,misalignment between the central axis of a plug for use in piercing andthe central axis of the billet, or uneven heating temperature of thebillet due to insufficient heating in the heating furnace, is listed. Afirst-order thickness deviation is caused by this factor. On the otherhand, a second-order thickness deviation and a fourth-order thicknessdeviation are caused by the misalignment of the roll position from a settarget position in the mandrel mill and the deviation of the radius of amandrel bar used in the mandrel mill from an appropriate value.Moreover, a third-order thickness deviation and a sixth-order thicknessdeviation are caused by tool defects in the reducing mill or the sizingmill. Further, as shown in FIG. 1B, there is a deviation of wallthickness changing in the longitudinal direction. A possible cause ofthis thickness deviation is a deviation due to inclined rolling, suchas, for example, the above-mentioned first-order thickness deviationcaused by the piercing mill.

In order to prevent occurrence of such thickness deviations, it isnecessary to measure the wall thickness of a manufactured seamless pipe,know the state of a thickness deviation, when it is detected, specifythe cause of the thickness deviation, and adjust the manufacturingequipment or manufacturing conditions to eliminate the specified cause.

Japanese Patent Application Laid-Open No. 59-7407 (1984) discloses atechnique of dividing the cause of a first-order thickness deviationthat occurs in an inclined rolling mill, such as a piercing mill, bynoticing the degree of twist of a thin portion in a cross section of aseamless pipe in the longitudinal direction. On the other hand, JapanesePatent Application Laid-Open No. 61-135409 (1986) discloses a techniqueof determining the cause of thickness deviation by measuring the wallthickness of a seamless pipe spirally and analyzing a first-orderthickness deviation, a third-order thickness deviation of a 120° cycle,and a second-order thickness deviation of a 180° cycle by Fourieranalysis of the measurement results. Further, Japanese PatentApplication Laid-Open No. 8-271241 (1996) discloses a technique in whichtwo gamma rays are passed through a seamless pipe, and a first-orderthickness deviation is specified based on the difference between theattenuation amounts of the two gamma rays.

In the prior arts mentioned above, the technique disclosed in JapanesePatent Application Laid-Open No. 59-7407 (1984) has the problem that,when there are various types of thickness deviations, it is difficult todetermine the form of twist of the first-order thickness deviation. Onthe other hand, since the technique disclosed in Japanese PatentApplication Laid-Open No. 61-135409 (1986) does not disclose a method ofanalyzing the position of a thickness deviation in the circumferentialdirection, this technique has the problem that it is impossible todetermine an adjustment position where manufacturing conditions are tobe adjusted to prevent the thickness deviation, and an adjustmentamount. Further, it does not disclose specific measures to prevent afourth-order thickness deviation of a 90° cycle, a sixth-order thicknessdeviation of a 60° cycle, etc. Additionally, the technique disclosed inJapanese Patent Application Laid-Open No. 8-271241 (1996) has theproblem that it is impossible to divide an even-number-order thicknessdeviation and an odd-number order thickness deviation.

BRIEF SUMMARY OF THE INVENTION

The present invention has been made with the aim of solving the aboveproblems, and it is an object of the present invention to provide amanufacturing method/manufacturing apparatus of seamless pipes, capableof preventing occurrence of thickness deviation by classifying the typeof thickness deviation by performing a complex Fourier transform onmeasured values of wall thickness measured at a plurality of points in across section in an axial direction of the pipe, calculating theposition of thickness deviation, from the phase of a complex Fouriercomponent, and adjusting the manufacturing conditions or manufacturingequipment of the pipe, and to provide a thickness deviation informationderivation apparatus for deriving information about thickness deviation,and a computer program for realizing the thickness deviation informationderivation apparatus by a computer.

More specifically, the object of the present invention is to provide amanufacturing method/manufacturing apparatus of seamless pipes, capableof preventing occurrence of thickness deviation by adjusting themanufacturing conditions or manufacturing equipment of the pipe,according to the type of thickness deviation, for the first-orderthickness deviation through sixth-order thickness deviation, and toprovide a thickness deviation information derivation apparatus forderiving information about a thickness deviation, and a computer programfor realizing the thickness deviation information derivation apparatusby a computer.

Further, it is another object of the present invention to provide amanufacturing method/manufacturing apparatus of seamless pipes, capableof preventing occurrence of thickness deviation by determining the formof twist of a thickness deviation in a longitudinal direction of thepipe by performing a complex Fourier transform in the longitudinaldirection on the thickness deviation amount and phase obtained for across section of the pipe in the axial direction, and adjusting themanufacturing conditions or manufacturing equipment of the pipeaccording to the form of twist, and to provide a thickness deviationinformation derivation apparatus for deriving information about athickness deviation, and a computer program for realizing the thicknessdeviation information derivation apparatus by a computer.

A manufacturing method of seamless pipes according to the firstinvention is a method of manufacturing seamless pipes while adjustingwall thickness based on measured values of wall thickness of a pipe, andcharacterized by comprising: a step of measuring the wall thickness at aplurality of points in a circumferential direction in a cross section ofthe pipe in an axial direction; a first calculation step of calculatinga complex Fourier component of each k-th-order thickness deviation ofthe wall thickness that changes periodically k times (k is a naturalnumber) in one turn by performing a complex Fourier transform in thecircumferential direction on a plurality of measured values of the wallthickness; a second calculation step of calculating a thicknessdeviation amount indicating a degree of thickness deviation of eachk-th-order thickness deviation, from an absolute value of the calculatedcomplex Fourier component; a third calculation step of calculating aposition of a thick portion or thin portion of each k-th-order thicknessdeviation, from a phase of the calculated complex Fourier component; andan adjustment step of adjusting the wall thickness of the pipe, based onthe thickness deviation amount and/or the position of the thick portionor thin portion, according to a method suited to each k-th-orderthickness deviation.

A manufacturing method of seamless pipes according to the secondinvention is characterized in that, in the first calculation step, areal part R(k) and an imaginary part I(k) of the complex Fouriercomponent of each k-th-order thickness deviation are calculated by${{R(k)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\cos\left( {\frac{2\pi}{N}{k\left( {i - 1} \right)}} \right)}} \right\}}}},{and}$${I(k)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\sin\left( {\frac{2\pi}{N}{k\left( {i - 1} \right)}} \right)}} \right\}}}$where N is the number of measurement points at which the wall thicknessis measured in the cross section in the axial direction, and WT(i) is ameasured value of the wall thickness at the i-th measurement point amongthe measurement points aligned in the circumferential direction, and, inthe second calculation step, a thickness deviation amount G(k) of eachk-th-order thickness deviation is calculated byG(k)=4√{square root over (R(k)² +I(k)²)}.

A manufacturing method of seamless pipes according to the thirdinvention is characterized in that, in the third calculation step, aposition argW(k) of the thick portion or a position argN(k) of the thinportion of each k-th-order thickness deviation is calculated with a unitof angle in which the position of a first measurement point is 0°, usingthe real part R(k) and imaginary part I(k) of the complex Fouriercomponent of each k-th-order thickness deviation, by${{{\arg\quad{W(k)}} = {\frac{1}{k}\left( {\tan^{- 1}\frac{I(k)}{R(k)}} \right)\frac{180}{\pi}}},{or}}\quad$${\arg\quad{N(k)}} = {\frac{1}{k}\left( {{\tan^{- 1}\frac{I(k)}{R(k)}} + \pi} \right){\frac{180}{\pi}.}}$

A manufacturing method of seamless pipes according to the fourthinvention is characterized in that a mandrel mill comprising a pluralityof rolling roll pairs for rolling a pipe by sandwiching a hollow shellfrom outside is used, and, in the adjustment step, for a second-orderthickness deviation, a distance between the rolling rolls of a rollingroll pair of the mandrel mill that roll the position of the thickportion is decreased according to the thickness deviation amount, or adistance between the rolling rolls of a rolling roll pair that roll theposition of the thin portion is increased according to the thicknessdeviation amount.

A manufacturing method of seamless pipes according to the fifthinvention is characterized in that a mandrel mill with a mandrel bar tobe inserted into a hollow shell is used, and, in the adjustment step,for a fourth-order thickness deviation, the mandrel bar of the mandrelmill is replaced by a mandrel bar having an appropriate diameteraccording to the thickness deviation amount.

A manufacturing method of seamless pipes according to the sixthinvention is characterized in that a reducing rolling mill for rolling apipe by passing the pipe through a die formed by a plurality of rollingrolls is used, and, in the adjustment step, for a third-order thicknessdeviation or a sixth-order thickness deviation, the rolls of thereducing rolling mill are replaced by rolls having an appropriate shape,based on the deviation amount and/or the position of the thick portionor thin portion.

A manufacturing method of seamless pipes according to the seventhinvention is a method of manufacturing seamless pipes while adjustingwall thickness based on measured values of wall thickness of a pipe, andcharacterized by comprising: a step of measuring the wall thickness at aplurality of points in a circumferential direction in a plurality ofcross sections of the pipe in an axial direction; a step of calculatinga complex Fourier component of each k-th-order thickness deviation ofwall thickness that changes periodically k times (k is a natural number)in one turn by performing a complex Fourier transform in thecircumferential direction on a plurality of measured values of the wallthickness in the plurality of the cross sections in the axial direction;a step of calculating a thickness deviation amount indicating a degreeof thickness deviation of each k-th-order thickness deviation, from anabsolute value of the calculated complex Fourier component, for theplurality of the cross sections in the axial direction; a step ofcalculating a phase of each k-th-order thickness deviation, from thecalculated complex Fourier component, for the plurality of the crosssections in the axial direction; a step of calculating a complex Fouriercomponent for each frequency representing the number of times thethickness deviation is twisted per unit length in a longitudinaldirection of the pipe, by performing a complex Fourier transform on acomplex function in which a complex number whose absolute value andphase are the thickness deviation and the phase calculated for eachk-th-order thickness deviation, respectively, is a function of positionin the longitudinal direction of the pipe; a step of determining whetherthe values of the complex Fourier component in a plurality ofpredetermined frequency ranges are larger or not based on apredetermined boundary value; and a step of adjusting the wall thicknessof the pipe, according to a method suited to each k-th-order thicknessdeviation and each frequency range, when it is determined that the valueof the complex Fourier component is larger in any of the frequencyranges.

A manufacturing method of seamless pipes according to the eighthinvention is a method of manufacturing seamless pipes while adjustingwall thickness based on measured values of wall thickness of a pipe, andcharacterized by comprising: a step of measuring the wall thickness at aplurality of points in a circumferential direction in a plurality ofcross sections of the pipe in an axial direction; a first calculationstep of calculating a complex Fourier component of a first-orderthickness deviation of the wall thickness that changes periodically oncein one turn by performing a complex Fourier transform in thecircumferential direction on a plurality of measured values of the wallthickness in the plurality of the cross sections in the axial direction;a second calculation step of calculating a thickness deviation amountindicating a degree of thickness deviation of the first-order thicknessdeviation, from an absolute value of the calculated complex Fouriercomponent, for the plurality of the cross sections in the axialdirection; a third calculation step of calculating a phase of thefirst-order thickness deviation, from the calculated complex Fouriercomponent, for the plurality of the cross sections in the axialdirection; a fourth calculation step of calculating a complex Fouriercomponent for each frequency representing the number of times thethickness deviation is twisted per unit length in a longitudinaldirection of the pipe, by performing a complex Fourier transform on acomplex function in which a complex number whose absolute value andphase are the thickness deviation amount and the phase thus calculated,respectively, is a function of position in the longitudinal direction ofthe pipe; a determination step of determining whether the values of thecomplex Fourier component in a plurality of predetermined frequencyranges are larger or not based on a predetermined boundary value; and anadjustment step of adjusting manufacturing conditions of the pipe, whenit is determined that the value of the complex Fourier component islarger in any of the frequency ranges, according to a method suited tothe frequency range.

A manufacturing method of seamless pipes according to the ninthinvention is characterized in that, in the first calculation step, areal part R(1) and an imaginary part I(1) of the complex Fouriercomponent of the first-order thickness deviation are calculated by${{R(1)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\cos\left( {\frac{2\pi}{N}\left( {i - 1} \right)} \right)}} \right\}}}},{and}$${I(1)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\sin\left( {\frac{2\pi}{N}1\left( {i - 1} \right)} \right)}} \right\}}}$where N is the number of measurement points at which the wall thicknessis measured in the cross section in the axial direction, and WT(i) is ameasured value of the wall thickness at the i-th measurement point amongthe measurement points aligned in the circumferential direction, and, inthe second calculation step, a thickness deviation amount r of thefirst-order thickness deviation is calculated byr=4√{square root over (R(1)² +I(1)²)},in the third calculation step, a phase θ of the first-order thicknessdeviation is calculated byθ=tan⁻¹ {I(1)/R(1)},and, in the fourth calculation step, a complex Fourier transform isperformed on a function of y, f(y)=r(y)·exp(j·θ(y)), where j is animaginary number, y is a length in the longitudinal direction of thepipe, and the r and θ are functions of y.

A manufacturing method of seamless pipes according to the tenthinvention is characterized in that a heating furnace and a piercing millare used; in the determination step, whether the value of the complexFourier component is larger or not is determined based on apredetermined boundary value, in a range of large frequencies and arange of small frequencies based on a predetermined boundary value; and,in the adjustment step, when it is determined that the value of thecomplex Fourier component is larger in the range of small frequencies, aheating temperature in the heating furnace is increased, and, when it isdetermined that the value of the complex Fourier component is larger inthe range of large frequencies, a part in the piercing mill that causeseccentricity is replaced.

A manufacturing apparatus according to the eleventh invention is anapparatus for manufacturing seamless pipes while adjusting wallthickness based on measured values of wall thickness of a pipe, andcharacterized by comprising: means for measuring the wall thickness at aplurality of points in a circumferential direction in a cross section ofthe pipe in an axial direction; first calculating means for calculatinga complex Fourier component of each k-th-order thickness deviation ofthe wall thickness that changes periodically k times (k is a naturalnumber) in one turn by performing a complex Fourier transform in thecircumferential direction on a plurality of measured values of the wallthickness; second calculating means for calculating a thicknessdeviation amount indicating a degree of thickness deviation of eachk-th-order thickness deviation, from an absolute value of the calculatedcomplex Fourier component; third calculating means for calculating aposition of a thick portion or thin portion of each k-th-order thicknessdeviation, from a phase of the calculated complex Fourier component; andadjusting means for adjusting the wall thickness of the pipe to bemanufactured, based on the thickness deviation amount and/or theposition of the thick portion or thin portion, according to a methodsuited to each k-th-order thickness deviation.

A manufacturing apparatus according to the twelfth invention ischaracterized in that the first calculating means calculates a real partR(k) and an imaginary part I(k) of the complex Fourier component of eachk-th-order thickness deviation by${{R(k)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\cos\left( {\frac{2\pi}{N}{k\left( {i - 1} \right)}} \right)}} \right\}}}},{and}$${I(k)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\sin\left( {\frac{2\pi}{N}{k\left( {i - 1} \right)}} \right)}} \right\}}}$where N is the number of measurement points at which the wall thicknessis measured in the cross section in the axial direction, and WT(i) is ameasured value of the wall thickness at the i-th measurement point amongthe measurement points aligned in the circumferential direction, and thesecond calculating means calculates a thickness deviation amount G(k) ofeach k-th-order thickness deviation byG(k)=4√{square root over (R(k)² +I(k))}.

A manufacturing apparatus according to the thirteenth invention ischaracterized in that the third calculating means calculates a positionargW(k) of the thick portion or a position argN(k) of the thin portionof each k-th-order thickness deviation with a unit of angle in which theposition of a first measurement point is 0°, using the real part R(k)and imaginary part I(k) of the complex Fourier component of eachk-th-order thickness deviation, by${{{\arg\quad{W(k)}} = {\frac{1}{k}\left( {\tan^{- 1}\frac{I(k)}{R(k)}} \right)\frac{180}{\pi}}},{or}}\quad$${\arg\quad{N(k)}} = {\frac{1}{k}\left( {{\tan^{- 1}\frac{I(k)}{R(k)}} + \pi} \right){\frac{180}{\pi}.}}$

A manufacturing apparatus according to the fourteenth invention is anapparatus for manufacturing seamless pipes while adjusting wallthickness based on measured values of wall thickness of a pipe, andcharacterized by comprising: means for measuring the wall thickness at aplurality of points in a circumferential direction in a plurality ofcross sections of the pipe in an axial direction; first calculatingmeans for calculating a complex Fourier component of a first-orderthickness deviation of the wall thickness that changes periodically oncein one turn by performing a complex Fourier transform in thecircumferential direction on a plurality of measured values of the wallthickness in the plurality of the cross sections in the axial direction;second calculating means for calculating a thickness deviation amountindicating a degree of thickness deviation of the first-order thicknessdeviation, from an absolute value of the calculated complex Fouriercomponent, for the plurality of the cross sections in the axialdirection; third calculating means for calculating a phase of thefirst-order thickness deviation, from the calculated complex Fouriercomponent, for the plurality of the cross sections in the axialdirection; fourth calculating means for calculating a complex Fouriercomponent for each frequency representing the number of times thethickness deviation is twisted per unit length in a longitudinaldirection of the pipe, by taking a relationship between the thicknessdeviation amount and the phase thus calculated as a function of positionin the longitudinal direction of the pipe and performing a complexFourier transform over a plurality of the functions; determining meansfor determining whether values of the complex Fourier component in aplurality of predetermined frequency ranges are larger or not based on apredetermined boundary value; and adjusting means for adjustingmanufacturing conditions of the pipe, when it is determined that thevalue of the complex Fourier component is larger in any of the frequencyranges, according to a method suited to the frequency range.

A manufacturing apparatus according to the fifteenth invention ischaracterized in that the first calculating means calculates a real partR(1) and an imaginary part I(1) of the complex Fourier component of thefirst-order thickness deviation by${{R(1)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\cos\left( {\frac{2\pi}{N}\left( {i - 1} \right)} \right)}} \right\}}}},{and}$${I(1)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\sin\left( {\frac{2\pi}{N}1\left( {i - 1} \right)} \right)}} \right\}}}$where N is the number of measurement points at which the wall thicknessis measured in the cross section in the axial direction, and WT(i) is ameasured value of the wall thickness at the i-th measurement point amongthe measurement points aligned in the circumferential direction, thesecond calculating means calculates a thickness deviation amount r ofthe first-order thickness deviation byr=4√{square root over (R(1)² +I(1)²)},the third calculating means calculates a phase θ of the first-orderthickness deviation byθ=tan⁻¹ {I(1)/R(1)},and the fourth calculating means performs a complex Fourier transform ona function of y, f(y)=r(y)·exp(j·θ(y)), where j is an imaginary number,y is a length in the longitudinal direction of the pipe, and the r and θare functions of y.

A thickness deviation information derivation apparatus according to thesixteenth invention is an apparatus for deriving information about athickness deviation that occurs in a pipe, based on measured values ofwall thickness of the pipe, and characterized by comprising: firstcalculating means for calculating a complex Fourier component of eachk-th-order thickness deviation of the wall thickness that changesperiodically k times (k is a natural number) in one turn by performing acomplex Fourier transform in a circumferential direction on measuredvalues of the wall thickness at a plurality of points in a cross sectionof the pipe in an axial direction; second calculating means forcalculating a thickness deviation amount indicating a degree ofthickness deviation of each k-th-order thickness deviation, from anabsolute value of the calculated complex Fourier component; and thirdcalculating means for calculating a position of a thick portion or thinportion of each k-th-order thickness deviation, from a phase of thecalculated complex Fourier component.

A thickness deviation information derivation apparatus according to theseventeenth invention is characterized in that the first calculatingmeans calculates a real part R(k) and an imaginary part I(k) of thecomplex Fourier component of each k-th-order thickness deviation by${{R(k)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\cos\left( {\frac{2\pi}{N}{k\left( {i - 1} \right)}} \right)}} \right\}}}},{and}$${I(k)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\sin\left( {\frac{2\pi}{N}{k\left( {i - 1} \right)}} \right)}} \right\}}}$where N is the number of measurement points at which the wall thicknessis measured in the cross section in the axial direction, and WT(i) is ameasured value of the wall thickness at the i-th measurement point amongthe measurement points aligned in the circumferential direction, and thesecond calculating means calculates a thickness deviation amount G(k) ofeach k-th-order thickness deviation byG(k)=4√{square root over (R(k)² +I(k)²)}.

A thickness deviation information derivation apparatus according to theeighteenth invention is characterized in that the third calculatingmeans calculates a position argW(k) of the thick portion or a positionargN(k) of the thin portion of each k-th-order thickness deviation witha unit of angle in which the position of a first measurement point is0°, using the real part R(k) and imaginary part I(k) of the complexFourier component of each k-th-order thickness deviation, by${{{\arg\quad{W(k)}} = {\frac{1}{k}\left( {\tan^{- 1}\frac{I(k)}{R(k)}} \right)\frac{180}{\pi}}},{or}}\quad$${\arg\quad{N(k)}} = {\frac{1}{k}\left( {{\tan^{- 1}\frac{I(k)}{R(k)}} + \pi} \right){\frac{180}{\pi}.}}$

A thickness deviation information derivation apparatus according to thenineteenth invention is an apparatus for deriving information about athickness deviation that occurs in a pipe, based on measured values ofwall thickness of the pipe, and characterized by comprising: firstcalculating means for calculating a complex Fourier component of afirst-order thickness deviation of the wall thickness that changesperiodically once in one turn by performing a complex Fourier transformin a circumferential direction on measured values of the wall thicknessat a plurality of points in a plurality of cross sections of the pipe inan axial direction; second calculating means for calculating a thicknessdeviation amount indicating a degree of thickness deviation of thefirst-order thickness deviation, from an absolute value of thecalculated complex Fourier component, for the plurality of the crosssections in the axial direction; third calculating means for calculatinga phase of the first-order thickness deviation, from the calculatedcomplex Fourier component, for the plurality of the cross sections inthe axial direction; fourth calculating means for calculating a complexFourier component for each frequency representing the number of timesthe thickness deviation is twisted per unit length in a longitudinaldirection of the pipe, by performing a complex Fourier transform on acomplex function in which a complex number whose absolute value andphase are the thickness deviation amount and the phase thus calculated,respectively, is a function of position in the longitudinal direction ofthe pipe; and determining means for determining whether the values ofthe complex Fourier component in a plurality of predetermined frequencyranges are larger or not based on a predetermined boundary value.

A thickness deviation information derivation apparatus according to thetwentieth invention is characterized in that the first calculating meanscalculates a real part R(1) and an imaginary part I(1) of the complexFourier component of the first-order thickness deviation by${{R(1)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\cos\left( {\frac{2\pi}{N}\left( {i - 1} \right)} \right)}} \right\}}}},{and}$${I(1)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\sin\left( {\frac{2\pi}{N}1\left( {i - 1} \right)} \right)}} \right\}}}$where N is the number of measurement points at which the wall thicknessis measured in the cross section in the axial direction, and WT(i) is ameasured value of the wall thickness at the i-th measurement point amongthe measurement points aligned in the circumferential direction, thesecond calculating means calculates a thickness deviation amount r ofthe first-order thickness deviation byr=4√{square root over (R(1)² +I(1)²)},the third calculating means calculates a phase θ of the first-orderthickness deviation byθ=tan⁻¹ {I(1)/R(1)},and the fourth calculating means performs a complex Fourier transform ona function of y, f(y)=r(y)·exp(j·θ(y)), where j is an imaginary number,y is a length in the longitudinal direction of the pipe, and the r and θare functions of y.

A computer program according to the twenty first invention is a computerprogram for causing a computer to derive information about a thicknessdeviation that occurs in a pipe, based on wall thickness values of thepipe, and characterized by comprising: a first calculation step ofcausing a computer to calculate a complex Fourier component of eachk-th-order thickness deviation of the wall thickness that changesperiodically k times (k is a natural number) by performing a complexFourier transform on a series of the wall thickness values of the pipe;a second calculation step of causing a computer to calculate a thicknessdeviation amount indicating a degree of thickness deviation of eachk-th-order thickness deviation, from an absolute value of the calculatedcomplex Fourier component; and a third calculation step of causing acomputer to calculate a position of a thick portion or thin portion ofeach k-th-order thickness deviation, from a phase of the calculatedcomplex Fourier component.

A computer program according to the twenty second invention ischaracterized in that the first calculation step includes a step ofcausing a computer to calculate a real part R(k) and an imaginary partI(k) of the complex Fourier component of each k-th-order thicknessdeviation by${{R(k)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\cos\left( {\frac{2\pi}{N}{k\left( {i - 1} \right)}} \right)}} \right\}}}},{and}$${I(k)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\sin\left( {\frac{2\pi}{N}{k\left( {i - 1} \right)}} \right)}} \right\}}}$where N is the number of the wall thickness values, and WT(i) is thei-th wall thickness value, and the second calculation step includes astep of causing a computer to calculate a thickness deviation amountG(k) of each k-th-order thickness deviation byG(k)=4√{square root over (R(k)² +I(k)²)}.

A computer program according to the twenty third invention ischaracterized in that the third calculation step includes a step ofcausing a computer to calculate a position argW(k) of the thick portionor a position argN(k) of the thin portion of each k-th-order thicknessdeviation with a unit of angle, using the real part R(k) and imaginarypart I(k) of the complex Fourier component of each k-th-order thicknessdeviation, by${{{\arg\quad{W(k)}} = {\frac{1}{k}\left( {\tan^{- 1}\frac{I(k)}{R(k)}} \right)\frac{180}{\pi}}},{or}}\quad$${\arg\quad{N(k)}} = {\frac{1}{k}\left( {{\tan^{- 1}\frac{I(k)}{R(k)}} + \pi} \right){\frac{180}{\pi}.}}$

A computer program according to the twenty fourth invention is acomputer program for causing a computer to derive information about athickness deviation that occurs in a pipe, based on wall thicknessvalues of the pipe, and characterized by comprising: a first calculationstep of causing a computer to calculate a complex Fourier component of afirst-order thickness deviation of the wall thickness that changesperiodically once by performing a complex Fourier transform on each of aplurality of series of the wall thickness values; a second calculationstep of causing a computer to calculate a thickness deviation amountindicating a degree of thickness deviation of the first-order thicknessdeviation, from an absolute value of the calculated complex Fouriercomponent, for each of a plurality of series of the wall thicknessvalues; a third calculation step of causing a computer to calculate aphase of the first-order thickness deviation, from the calculatedcomplex Fourier component, for each of a plurality of series of the wallthickness values; a fourth calculation step of causing a computer tocalculate a complex Fourier component for each frequency by performing acomplex Fourier transform on a relationship between the thicknessdeviation amount and the phase calculated for each of a plurality ofseries of the wall thickness values, over a plurality of series of thewall thickness values; and a step of causing a computer to determinewhether values of the complex Fourier component in a plurality ofpredetermined frequency ranges are larger or not based on apredetermined boundary value.

A computer program according to the twenty fifth invention ischaracterized in that the first calculation step includes a step ofcausing a computer to calculate a real part R(1) and an imaginary partI(1) of the complex Fourier component of the first-order thicknessdeviation by${{R(1)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\cos\left( {\frac{2\pi}{N}\left( {i - 1} \right)} \right)}} \right\}}}},{and}$${I(1)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\sin\left( {\frac{2\pi}{N}1\left( {i - 1} \right)} \right)}} \right\}}}$where N is the number of the wall thickness values included in a seriesof the wall thickness values, and WT(i) is the i-th wall thickness valueincluded in the series of the wall thickness values, the secondcalculation step includes a step of causing a computer to calculate athickness deviation amount r of the first-order thickness deviation byr=4√{square root over (R(1)² +I(1)²)},the third calculation step includes a step of causing a computer tocalculate a phase θ of the first-order thickness deviation byθ=tan⁻¹ {I(1)/R(1)},and the fourth calculation step includes a step of causing a computer toperform complex Fourier transform on a function of y,f(y)=r(y)·exp(j·θ(y)), where j is an imaginary number, y is a length ina longitudinal direction of the pipe, and the r and θ are functions ofy.

In the first, eleventh, sixteenth, and twenty first invention, a complexFourier transform is performed on measured values of the wall thicknessat a plurality of points in a cross section of a seamless pipe in theaxial direction, the type of thickness deviation is classified, athickness deviation amount is calculated from the absolute value of acomplex Fourier component, the position of a thick portion or thinportion of the thickness deviation is calculated from the phase of thecomplex Fourier component, and manufacturing conditions of the pipe areadjusted based on the type of thickness deviation, the thicknessdeviation amount, and the position of the thick portion or thin portion.Since the absolute value of the complex Fourier component gives theamplitude of each k-th-order thickness deviation of the wall thicknessthat changes periodically k times in one turn, the thickness deviationamount of each k-th-order thickness deviation is obtained. Further,since the phase given by the ratio between the real part and theimaginary part of the complex Fourier component indicates the degree ofdeviation of a curve drawn from the measurement start point as theorigin by plotting the angle in the circumferential direction of thepipe on the abscissa and the change of the k-th-order thicknessdeviation on the ordinate, from a cosine curve composed only of the realpart, the position of the thick portion where the curve has a maximumvalue and the position of the thin portion where the curve has a minimumvalue are obtained from the phase. Consequently, the type of thicknessdeviation, the thickness deviation amount, and the position of the thickportion or thin portion are found, and an appropriate action to preventthe thickness deviation is taken.

In the second, twelfth, seventeenth, and twenty second invention, whenthe thickness deviation amount of each k-th-order thickness deviation isdefined as the value obtained by subtracting the minimum wall thicknessfrom the maximum wall thickness, then the thickness deviation amount is4√{square root over (R(k)²+I(k)²)} which is twice the amplitude 2√{square root over (R(k) ² +I(k) ² )} of a sine wave drawn by eachk-th-order thickness deviation, and, since the thickness deviationamount of each k-th-order thickness deviation is calculated, anadjustment amount for adjusting the wall thickness of the pipe toprevent the thickness deviation is determined.

In the third, thirteenth, eighteenth, and twenty third invention, theposition of the thick portion of each k-th-order thickness deviation isthe position of an angle obtained by dividing the phase given by theratio between the real part and the imaginary part of the complexFourier component by k, and the position of the thin portion is theposition of an angle obtained by dividing, by k, the value obtained byadding two right angles to the phase indicating the position of thethick portion within one cycle, and, since the position of the thickportion or thin portion of each k-th-order thickness deviation iscalculated, a position where the wall thickness of the pipe is to beadjusted to prevent the thickness deviation and whether the adjustmentamount is positive or negative are specified.

In the fourth invention, when a second-order thickness deviation occurs,the manufacturing conditions of the pipe are adjusted to eliminate thecause of the second-order thickness deviation. The second-orderthickness deviation is caused by a variation of pressure during rollingby the mandrel mill, and therefore, when the second-order thicknessdeviation occurs, the distance between the rolling rolls of a rollingroll pair rolling the position of the thick portion is decreasedaccording to the deviation amount, and the distance between the rollingrolls of a rolling pair rolling the position of the thin portion isincreased according to the thickness deviation amount.

In the fifth invention, when a fourth-order thickness deviation occurs,the manufacturing conditions of the pipe are adjusted to eliminate thecause of the fourth-order thickness deviation. For the fourth-orderthickness deviation, the present inventors found out that the thicknessdeviation amount can be changed by changing the shape of the mandrelbar, and therefore, when the fourth-order thickness deviation occurs,the mandrel bar is replaced by a bar having a different diameteraccording to the thickness deviation amount.

In the sixth invention, when a third-order thickness deviation and asixth-order thickness deviation occur, the manufacturing conditions ofthe pipe are adjusted to eliminate the cause of the thicknessdeviations. For the third-order thickness deviation and sixth-orderthickness deviation, the present inventors found out that the thicknessdeviation amount can be reduced by adjusting the shape of the die of thereducing rolling mill that is a reducing mill or a sizing mill, andtherefore, when the third-order thickness deviation and sixth-orderthickness deviation occur, the rolls are replaced according to thethickness deviation amount and the position of the thick portion or thinportion.

In the seventh invention, the relationship between the thicknessdeviation amount and the phase of each k-th-order thickness deviationobtained for a plurality of cross sections in the axial direction istaken as a function of the longitudinal direction of the pipe, a complexFourier transform is performed on the function, the thickness deviationis further classified by the frequency of twist of the thicknessdeviation, and an appropriate action is taken to prevent the thicknessdeviation according to the classified thickness deviation.

In the eighth, fourteenth, nineteenth, and twenty fourth invention, therelationship between the thickness deviation amount and phase of afirst-order thickness deviation obtained for a plurality of crosssections in the axial direction is taken as a function of thelongitudinal direction of the pipe, a complex Fourier transform isperformed on the function, the thickness deviation is further classifiedby the frequency of twist of the thickness deviation, and an appropriateaction is taken to prevent the thickness deviation according to theclassified thickness deviation.

In the ninth, fifteenth, twentieth, and twenty fifth invention, for thethickness deviation amount r and phase θ of a first-order thicknessdeviation obtained for a plurality of cross sections in the axialdirection, a complex Fourier transform is performed on a function of y,f(y)=r(y)·exp(j·θ(y)), where j is an imaginary number, y is a length inthe longitudinal direction of the pipe, and the r and θ are functions ofy. When m is the frequency of twist and αm is an angular frequencycorresponding to the frequency m of twist, then the frequency m of twistof the first-order thickness deviation is analyzed by a result ofcomplex Fourier transform corresponding to a component of exp(j(αm·y)),and an appropriate action is taken to prevent a thickness deviationaccording to the thickness deviation classified by the frequency.

In the tenth invention, when a first-order thickness deviation with alarge frequency of twist occurs, and when a first-order thicknessdeviation with a small frequency of twist occurs, the manufacturingconditions of the pipe are adjusted to eliminate the causes of therespective thickness deviations. When the frequency of twist is small,i.e., when the first-order thickness deviation is twisted over a longlength in the longitudinal direction of the pipe, the cause of thethickness deviation is a variation of heating in the heating furnace,and therefore the temperature of heating the billet is increased byincreasing the heating time or by increasing the temperature in theheating furnace. When the frequency of twist is large, i.e., when thefirst-order thickness deviation is twisted in a short length in thelongitudinal direction of the pipe, the cause of the thickness deviationis misalignment between the central axis of a tool used for piercing andthe central axis of the billet, and therefore the part that causeseccentricity is replaced.

The above and further objects and features of the invention will morefully be apparent from the following detailed description withaccompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A and FIG. 1B are cross sectional views of a seamless pipe withvarious types of thickness deviations;

FIG. 2 is a block diagram showing a manufacturing apparatus of seamlesspipe of the present invention;

FIG. 3 is a block diagram showing the configuration of a thicknessdeviation information derivation apparatus of the present invention;

FIG. 4 is a schematic view showing the configuration of a mandrel mill;

FIG. 5A and FIG 5B are schematic views showing the configuration of areducing mill;

FIG. 6 is a schematic cross sectional view in the axial direction,showing an example of the configuration of a wall thickness meter;

FIG. 7 is a flowchart showing the procedure of the process performed bythe thickness deviation information derivation apparatus of the presentinvention;

FIG. 8 is a flowchart showing the procedure of the process performed bythe thickness deviation information derivation apparatus of the presentinvention;

FIG. 9 is a characteristic view showing a complex function f(y); and

FIG. 10 is a table showing an example of the result of application ofthe present invention.

DETAILED DESCRIPTION OF THE PRESENT INVENTION

The following description will specifically explain the presentinvention, based on the drawings illustrating some embodiments thereof.

FIG. 2 is a block diagram showing a manufacturing apparatus of seamlesspipes of the present invention. A seamless pipe is manufactured by theprocess of heating a billet to a required temperature in a heatingfurnace 4, piercing and rolling the billet by a piercing mill 5 toobtain a hollow shell, drawing and rolling the hollow shell using amandrel mill 6, and performing sizing rolling using a reducing mill foradjusting the shape, such as the external diameter and wall thickness.The manufacturing apparatus of the present invention comprises: a wallthickness meter 21 provided on the exit side of the mandrel mill 6; awall thickness meter 22 provided on the exit side of the reducing mill7; a thickness deviation information derivation apparatus 1 to which thewall thickness meters 21 and 22 are connected; an output device 3,connected to the thickness deviation information derivation apparatus 1,for outputting information from the thickness deviation informationderivation apparatus 1; a controller 41, connected to the thicknessdeviation information derivation apparatus 1, for controlling theheating furnace 4 based on the information from the thickness deviationinformation derivation apparatus 1; and a controller 61, connected tothe thickness deviation information derivation apparatus 1, forcontrolling the mandrel mill 6 based on the information from thethickness deviation information derivation apparatus 1.

FIG. 3 is a block diagram showing the configuration of the thicknessdeviation information derivation apparatus 1 of the present invention.The thickness deviation information derivation apparatus 1 isconstructed using a computer, comprises: a CPU 11 for performingoperations; a RAM 12 for storing temporary information generated by theoperations; an external storage device 13 such as a CD-ROM drive; and aninternal storage device 14 such as a hard disk or a semiconductormemory, reads a computer program 100 of the present invention from amemory product 10 such as a CD-ROM by the external storage device 13,stores the read computer program 100 into the internal storage device14, loads the computer program 100 into the RAM 12, and executesprocessing necessary for the thickness deviation information derivationapparatus 1, based on the loaded computer program 100. The thicknessdeviation information derivation apparatus 1 comprises an input unit 15connected to the wall thickness meters 21 and 22, and receivesinformation from the wall thickness meters 21 and 22 through the inputunit 15. Moreover, the thickness information derivation apparatus 1comprises an output unit 16 connected to the output device 3 and thecontrollers 41 and 61, and outputs information to the output device 3and the controllers 41 and 61 through the output unit 16.

FIG. 4 is a schematic view showing the configuration of the mandrel mill6. The mandrel mill 6 comprises: a first stand 63 including a pair ofrolling rolls 65, 65 for rolling a pipe P from the right and left sidesand a second stand 64 including a pair of rolling rollers 65, 65 forrolling the pipe P from the upper and lower sides, which are arrangedalternately; and a mandrel bar 62 to be inserted into the pipe P, androlls the pipe P into which the mandrel bar 62 is inserted, from theupper, lower, right and left sides. Each of the first stands 63, 63, . .. and the second stands 64, 64, . . . has a rolling adjustor 66 foradjusting the distance between a pair of rolling rolls 65, 65, and eachof the rolling adjustors 66, 66, . . . is connected to the controller61. The rolling adjustors 66, 66, . . . are controlled by the controller61, and adjust the distance between the rolling rolls 65 and 65.

FIG. 5A and FIG. 5B are schematic views showing the configuration of thereducing mill 7. FIG. 5A is a perspective view of the reducing mill 7.The reducing mill 7 comprises stands 71, 71, . . . along a longitudinaldirection of the pipe P, and each stand 71 has a set of threebobbin-like rolls 72, 72, 72. FIG. 5B is a front view of the stand 71. Adie is formed by a set of three rolls 72, 72, 72, and the reducing mill7 adjusts the shape of the pipe P by inserting the pipe P into the dieand rolling the pipe P.

FIG. 6 is a schematic cross sectional view in the axial direction,showing an example of the configuration of the wall thickness meter 21.In FIG. 6, the structure of an example of measuring the wall thicknessof the pipe P at 9 points on the periphery of the pipe P by usingradiation is illustrated, and 211 a through 219 a in FIG. 6 areradiation sources, and 2111 b through 219 b in FIG. 6 are radiationdetectors. Radiation from the radiation source 211 a is detected by theradiation detector 211 b, radiation from the radiation source 212 a isdetected by the radiation detector 212 b, and detection of radiation iscarried out in the same manner by the other radiation detectors. Thelength that the radiation passed through the pipe P is measured from theattenuation of radiation detected by the radiation detector. The pointof 211 shown in FIG. 6 is measured by the radiation source 211 a and theradiation detector 211 b, and the radiation source 215 a and theradiation detector 215 b, and each of other points is also measuredtwice in the same manner. Moreover, the length obtained by the sum ofthe wall thicknesses at the point of 211 and the point of 216 shown inFIG. 6 is measured from the radiation detected by the radiation detector211 b, and similarly the length obtained by the sum of the wallthicknesses at two points is also measured for each of the otherradiation detectors. By solving a simultaneous equation where the sum ofthe wall thicknesses at two points is equal to a length measured by eachradiation detector, the wall thicknesses at the points 211 through 219are measured. The wall thickness meter 22 also has the sameconfiguration.

Next, referring to the flowchart, the following description will explaina manufacturing method of seamless pipes of the present invention. FIG.7 and FIG. 8 are a flowchart showing the procedure of the processperformed by the thickness deviation information derivation apparatus 1of the present invention. The wall thickness meters 21 and 22 measurethe wall thickness of a pipe P to be manufactured, at a plurality ofpoints in the cross section in the axial direction at a constant pitchin the longitudinal direction, and input the measured values of wallthickness to the thickness deviation information derivation apparatus 1.The thickness deviation information derivation apparatus 1 receives themeasured values of wall thickness from the wall thickness meters 21 and22 at the input unit 15 (S1), and the CPU 11 of the thickness deviationinformation derivation apparatus 1 loads the computer program 100 intothe RAM 12, performs a complex Fourier transform on the receivedmeasured values and calculates a complex Fourier component of eachk-th-order thickness deviation of wall thickness that changesperiodically k times in one turn, according to the loaded computerprogram 100 (S2). Here, by representing the number of measurement pointsat which the wall thickness is measured in the cross section by N andrepresenting the measured value of the wall thickness at the i-thmeasurement point among measurement points aligned in a circumferentialdirection of the cross section by WT(i), a real part R (k) and animaginary part I(k) of the complex Fourier component of each k-th-orderthickness deviation are calculated from the discretely obtained WT(i) by${{R(k)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\cos\left( {\frac{2\pi}{N}{k\left( {i - 1} \right)}} \right)}} \right\}}}},{and}$${I(k)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\sin\left( {\frac{2\pi}{N}{k\left( {i - 1} \right)}} \right)}} \right\}}}$

Next, according to the computer program 100 loaded into the RAM 12, theCPU 11 calculates a thickness deviation amount G(k) of each k-th-orderthickness deviation (S3) byG(k)=4√{square root over (R(k)² +I(k)²)}.When the thickness deviation amount G(k) is defined as the valueobtained by subtracting the minimum wall thickness from the maximum wallthickness, then the thickness deviation amount G(k) is 4√{square rootover (R(k)²+I(k)²)}, which is twice the amplitude 2√{square root over(R(k)²+I(k)²)} of a sine wave drawn by each k-th-order thicknessdeviation.

Next, according to the computer program 100 loaded into the RAM 12, theCPU 11 calculates the phase of the complex Fourier component of eachk-th-order thickness deviation and the positions of a thick portion anda thin portion (S4). A phase θ of the complex Fourier component isobtained from the ratio between the real part R(k) and the imaginarypart I(k) of the complex Fourier component, and here the CPU 11calculates the phase θ by θ=tan⁻¹{I(k)/R(k)}. Moreover, since the phaseθ indicates the degree of deviation of a curve drawn from themeasurement start point i=1 as the origin by plotting the angle in thecircumferential direction of the pipe P on the abscissa and the changeof the k-th-order thickness deviation on the ordinate, from a cosinecurve composed only of the real part, here the CPU 11 calculates oneposition argW(k) of thick portion where the above-mentioned curve has amaximum value, with a unit of angle in which the position of themeasurement point i=1 is 0°, by${\arg\quad{W(k)}} = {\frac{1}{k}\left( {\tan^{- 1}\frac{I(k)}{R(k)}} \right){\frac{180}{\pi}.}}$Moreover, the positions argW(k) of k thick portions of the k-th-orderthickness deviation are given by${{\arg W}(k)} = {\frac{1}{k}\left( {{\tan^{- 1}\frac{I(k)}{R(k)}} + {2\left( {n - 1} \right)\pi}} \right)\frac{180}{\pi}}$where n is a natural number, n=1, 2, . . . , k. Further, since theposition of the thin portion is the position obtained by adding tworight angles to the position of the thick portion within one cycle ofchange of the thickness deviation, here the CPU 11 calculates oneposition argN(k) of thin portion, or the positions argN(k) of k thinportions of the k-th-order thickness deviation as follows.${\arg\quad{N(k)}\frac{1}{k}\left( {{\tan^{- 1}\frac{I(k)}{R(k)}} + \pi} \right)\frac{180}{\pi}},{or}$${\arg\quad{N(k)}} = {\frac{1}{k}\left( {{\tan^{- 1}\frac{I(k)}{R(k)}} + {\left( {{2n} - 1} \right)\pi}} \right)\frac{180}{\pi}}$

Next, according to the computer program 100 loaded into the RAM 12, theCPU 11 determines whether or not the calculated thickness deviationamount G(2) of a second-order thickness deviation exceeds apredetermined boundary value, i.e., whether or not there is asecond-order thickness deviation exceeding the tolerance (S5). Whenthere is a second-order thickness deviation (S5: YES), according to thecomputer program 100 loaded into the RAM 12, the CPU 11 outputsinformation for preventing the second-order thickness deviation to thecontroller 61 of the mandrel mill from the output unit 16, and controlsthe mandrel mill (S6). At this time, the thickness deviation informationderivation apparatus 1 outputs information including the calculatedthickness deviation amount G(2), the position argW(2) of the thickportion and the position argN(n) of the thin portion, and, according tothe information outputted from the thickness deviation informationderivation apparatus 1, the controller 61 controls the mandrel mill toprevent the second-order thickness deviation by causing the rollingadjustor 66 of a stand comprising a pair of rolling rolls 65, 65 rollingin the direction of the thick portion to decrease the distance betweenthe rolling rolls 65 and 65, and causing the rolling adjustor 66 of astand comprising a pair of rolling rolls 65, 65 rolling in the directionof the thin portion to increase the distance between the rolling rolls65 and 65.

When step S6 is completed, and when there is no second-order thicknessdeviation in step S5 (S5: NO), according to the computer program 100loaded into the RAM 12, the CPU 11 determines whether or not thecalculated thickness deviation amount G(4) of a fourth-order thicknessdeviation exceeds a predetermined boundary value, i.e., whether or notthere is a fourth-order thickness deviation exceeding the tolerance(S7), and, when there is a fourth-order thickness deviation (S7: YES),the CPU 11 outputs, from the output unit 16 to the output device 3, aninstruction to replace the mandrel bar 61 of the mandrel mill 6 by amandrel bar 61 having an appropriate diameter according to the thicknessdeviation amount (S8). The instruction to replace the mandrel bar 61 isoutputted to the output device 3, and the mandrel bar 61 is replaced byanother mandrel bar 61 having an appropriate diameter that can preventthe fourth-order thickness deviation by the work of an operator.

When step S8 is completed, and when there is no fourth-order thicknessdeviation in step S7 (S7: NO), according to the computer program 100loaded into the RAM 12, the CPU 11 determines whether or not thecalculated thickness deviation amount G(3) of a third-order thicknessdeviation or thickness deviation amount G(6) of a sixth-order thicknessdeviation exceeds a predetermined boundary value, i.e., whether or notthere is a third-order thickness deviation or a sixth-order thicknessdeviation exceeding the tolerance (S9), and, when there is a third-orderthickness deviation or a sixth-order thickness deviation (S9: YES), theCPU 11 outputs an instruction to adjust the shape of the die of thereducing mill 7 to the output device 3 from the output unit 16 (S10).The instruction to adjust the shape of the die is outputted to theoutput device 3, and the rolls 72, . . . forming the die that causes thethickness deviation are replaced by new rolls 72, . . . according to thethickness deviation amount, the position of the thick portion and theposition of the thin portion so as to prevent the third-order thicknessdeviation or the sixth-order thickness deviation.

When step S10 is completed, and when there is no third-order thicknessdeviation or sixth-order thickness deviation in step S9 (S9: NO),according to the computer program 100 loaded into the RAM 12, the CPU 11performs, based on the thickness deviation amount G(1) of thefirst-order thickness deviation obtained for each cross section in theaxial direction and the phase θ of the first-order thickness deviation,a complex Fourier transform in the y direction, namely, the longitudinaldirection of the pipe, on a function of y, f(y)=r(y)·exp(j·θ(y)), wherej is an imaginary number, y is a length in the longitudinal direction ofthe pipe, and the r and θ are functions of y, and calculates a complexFourier component for each frequency m by supposing that m is afrequency of twist, αm is an angular frequency corresponding to thefrequency m of twist, and θ=αm·y (S11). FIG. 9 is a characteristic viewshowing a complex function f(y). r(y)·exp(jθ) is represented as a pointof polar coordinates with R(1)/4 as the abscissa and I(1)/4 as theordinate, and, when the first-order thickness deviation is twisted inthe longitudinal direction of the pipe, the complex function f(y) drawsa substantially spiral trace by connecting the points of polarcoordinates in the y direction. By performing the complex Fouriertransform on f(y), it is possible to analyze the frequency m of twist ofthe first-order thickness deviation.

Next, according to the computer program 100 loaded into the RAM 12, theCPU 11 determines whether or not the absolute value of the complexFourier component exceeds a predetermined boundary value in a range ofnot more than a predetermined frequency value m=m0, i.e., whether or notthere is a twisted first-order thickness deviation in a range where thefrequency m is small (S12), and, when there is such a first-orderthickness deviation (S12: YES), the CPU 11 outputs information forpreventing the first-order thickness deviation twisted over a longlength in the longitudinal direction of the pipe P to the controller 41of the heating furnace 4, and controls the heating furnace 4 (S13). Atthis time, the thickness deviation information derivation apparatus 1outputs information including the absolute value of the calculatedcomplex Fourier component, and the controller 41 increases thetemperature of heating the billet by increasing the heating time of thebillet in the heating furnace 4, or by adjusting the heating power toincrease the temperature in the heating furnace 4, according to theinformation outputted from the thickness deviation informationderivation apparatus 1, thereby reducing the occurrence of thicknessdeviation due to uneven heating.

When step S13 is completed, and when there is no twisted first-orderthickness deviation in a range where the frequency m is small in stepS12 (S12: NO), according to the computer program 100 loaded into the RAM12, the CPU 11 determines whether or not the absolute value of thecomplex Fourier component exceeds a predetermined boundary value in arange exceeding a predetermined frequency value m=m0, i.e., whether ornot there is a twisted first-order thickness in a range where thefrequency m is large (S14), and, when there is such a first-orderthickness deviation (S14: YES), the CPU 11 outputs an instruction toreplace the tool that causes eccentricity in the piercing mill 5 to theoutput device 3 from the output unit 16 (S15). The instruction toreplace the tool is outputted to the output device 3, and the toolcausing eccentricity in the piercing mill 5 is replaced by the work ofthe operator to prevent the thickness deviation. When step S15 iscompleted, and when there is no twisted first-order thickness deviationin a range where the frequency m is large in step S14 (S14: NO), the CPU11 completes the processing.

As described in detail above, in the present invention, the complexFourier transform is performed on the measured values of wall thicknessat a plurality of points in a cross section of the pipe in the axialdirection, the thickness deviation amount and the positions of a thickportion and a thin portion are calculated from the calculated complexFourier component for each k-th-order thickness deviation, the positionof adjusting manufacturing conditions of the pipe to prevent a thicknessdeviation and whether the adjustment amount is positive or negative arespecified in addition to the type of thickness deviation, and anappropriate action can be taken to prevent the thickness deviation.Furthermore, by performing the complex Fourier transform in thelongitudinal direction of the pipe on the relationship between thethickness deviation amount and the phase, it is possible to classify thethickness deviation by the frequency of twist, and take an appropriateaction to prevent the thickness deviation according to the cause of thethickness deviation.

Next, an example of the result of application of the present inventionto the manufacturing process of seamless pipes is illustrated. FIG. 10is a table showing the example of the result of application of thepresent invention. The contents of processes carried out are as follows.

(a) Replacing a tool in the piercing mill 5 when there is a twistedfirst-order thickness deviation in the range of large frequencies.

(b) Increasing the heating time in the heating furnace 4 when there is atwisted first-order thickness deviation in the range of smallfrequencies.

(c) Adjusting the mandrel mill 6 when there is a second-order thicknessdeviation.

(d) Replacing the mandrel bar 61 when there is a fourth-order thicknessdeviation.

(e) Replacing the rolls of the reducing mill 7 when there is athird-order thickness deviation or a sixth-order thickness deviation.

(f) Taking actions against thickness deviations by the method disclosedin Japanese Patent Application Laid-Open No. 61-135409 (1986).

(g) No action.

When the present invention was carried out according to theabove-mentioned contents, the thickness deviation rate defined asthickness deviation rate=(thickness deviation amount/averagethickness)×100 was calculated based on the data measured by the wallthickness meter 22, and the ratio of the number of seamless pipes withthe thickness deviation rate equal to or more than 12.5% is shown inFIG. 10. As shown in FIG. 10, by using the present invention, the ratioof seamless pipes in which thickness deviations occurred is reducedcompared to the case where no action was taken against thicknessdeviations. Furthermore, the ratio of seamless pipes in which thicknessdeviation occurred is reduced compared to prior arts, and thus it isclear that the present invention has a superior ability to reduce theoccurrence of thickness deviations compared to the prior arts. Thus,with the present invention, it is possible to reduce the occurrence ofvarious types of thickness deviations during the manufacture of seamlesspipes and improve the quality of the seamless pipes.

Note that this embodiment illustrates a configuration in which the wallthickness meters 21 and 22 are provided on the exit side of the mandrelmill 6 and the exit side of the reducing mill 7, and the wall thicknessmeters 21 and 22 are connected to a common thickness deviationinformation derivation apparatus 1, but the present invention is notnecessarily limited to this configuration and may be implemented in aconfiguration in which two thickness deviation information derivationapparatuses 1 are provided and the wall thickness meters 21 and 22 areindividually connected to the respective thickness deviation informationderivation apparatuses 1, or a configuration including only the wallthickness meter 22 on the exit side of the reducing mill 7 as the wallthickness meter.

Besides, this embodiment illustrates a configuration in which thethickness deviation information derivation apparatus 1 is constructedusing a computer and performs processing related to the presentinvention according to the computer program 100, but the presentinvention is not limited to this configuration, and may be implementedin a configuration in which the thickness deviation informationderivation apparatus 1 of the present invention is constructed byhardware for exclusive use, such as storing means for storinginformation from the wall thickness meter and converting means forperforming a complex Fourier transform on measured values of wallthickness.

Moreover, although this embodiment illustrates a configuration using amethod in which rolling is performed using a mandrel mill and the sizeis adjusted using the reducing mill during the manufacture of seamlesspipes, the present invention is also applicable to a method ofmanufacturing seamless pipes by using rolling means other than a mandrelmill, such as a plug mill, and a reducing rolling mill other than areducing mill, such as a sizing mill.

Furthermore, although this embodiment illustrates a method of analyzingthe twist by performing a complex Fourier transform in the longitudinaldirection only on the first-order thickness deviation, it may also bepossible to use a method in which the complex Fourier transform isperformed in the longitudinal direction on other k-th-order thicknessdeviations, the twist is analyzed, and actions are taken against therespective thickness deviations.

INDUSTRIAL APPLICABILITY

As described in detail above, in the present invention, it is possibleto prevent a thickness deviation by performing a complex Fouriertransform on measured values of wall thickness at a plurality of pointsin a cross section of a pipe in the axial direction, classifying thetype of thickness deviation, calculating a thickness deviation amountfrom an absolute value of a complex Fourier component, calculating theposition of a thick portion or thin portion of the thickness deviation,from the phase of the complex Fourier component, and adjustingmanufacturing conditions of the pipe, based on the type of thicknessdeviation, the thickness deviation amount and the position of the thickportion or thin portion.

Moreover, in the present invention, an adjustment amount for adjustingthe wall thickness of the pipe to prevent a thickness deviation isdetermined by calculating the thickness deviation amount of eachk-th-order thickness deviation, and the wall thickness of the pipe canbe appropriately adjusted to prevent the thickness deviation.

Furthermore, in the present invention, the position where the wallthickness of the pipe is to be adjusted to prevent a thickness deviationand whether the adjustment amount is positive or negative are specifiedby calculating the position of a thick portion or thin portion of eachk-th-order thickness deviation, and the wall thickness of the pipe canbe appropriately adjusted.

Additionally, in the present invention, when a second-order thicknessdeviation occurs, the rolling condition of the mandrel mill is adjustedaccording to the thickness deviation amount and the position of thethickness deviation, thereby reducing the occurrence of a second-orderthickness deviation and improving the quality of the seamless pipe to bemanufactured.

Further, in the present invention, when a fourth-order thicknessdeviation occurs, the mandrel bar is replaced by another bar of adifferent diameter according to the thickness deviation amount, therebyreducing the occurrence of a fourth-order thickness deviation andimproving the quality of the seamless pipe to be manufactured.

Besides, in the present invention, when a third-order thicknessdeviation and a sixth-order thickness deviation occur, the rolls of thereducing rolling mill are replaced according to the thickness deviationamount and the position of the thickness deviation, thereby reducing theoccurrence of a third-order thickness deviation and a sixth-orderthickness deviation and improving the quality of the seamless pipe to bemanufactured.

Moreover, in the present invention, the relationship between thethickness deviation amount and the phase of each k-th-order thicknessdeviation obtained for a plurality of cross sections in the axialdirection is taken as a function of the longitudinal direction of thepipe, a complex Fourier transform is performed on the function, thethickness deviation is further classified by the frequency of twist ofthe thickness deviation, and the wall thickness of the pipe can beappropriately adjusted to prevent a thickness deviation, according tothe classified thickness deviation.

Furthermore, in the present invention, the relationship between thethickness deviation amount and the phase of a first-order thicknessdeviation obtained for a plurality of cross sections in the axialdirection is taken as a function of the longitudinal direction of thepipe, a complex Fourier transform is performed on the function, thethickness deviation is further classified by the frequency of twist ofthe thickness deviation, and the wall thickness of the pipe can beappropriately adjusted to prevent a thickness deviation, according tothe classified thickness deviation.

Additionally, in the present invention, a function r(y)·exp(jθ) ofthickness deviation amount r and phase θ of a first-order thicknessdeviation obtained for a plurality of cross sections in the axialdirection is taken as a function of the longitudinal direction of thepipe, a complex Fourier transform is performed on the function, thefrequency of twist of the first-order thickness deviation is analyzed,and the wall thickness of the pipe can be appropriately adjustedaccording to a thickness deviation classified by the frequency.

Besides, in the present invention, when a first-order thicknessdeviation with a small frequency of twist occurs, the heatingtemperature of the billet is increased, whereas when a first-orderthickness deviation with a large frequency of twist occurs, the toolthat causes eccentricity in the piercing mill is replaced, andconsequently the present invention provides advantages effects such asreducing the occurrence of the respective first-order thicknessdeviations and improving the quality of the seamless pipe to bemanufactured.

As this invention may be embodied in several forms without departingfrom the spirit of essential characteristics thereof, the presentembodiments are therefore illustrative and not restrictive, since thescope of the invention is and all changes that fall within metes andbounds of the claims, or equivalence of such meters and bounds thereofare therefore intended to be embraced by the claims.

1-28. (canceled)
 29. A thickness deviation information derivationapparatus for deriving information about a thickness deviation thatoccurs in a pipe, based on measured values of wall thickness of thepipe, comprising: first calculating means for calculating a complexFourier component of each k-th-order thickness deviation of the wallthickness that changes periodically k times (k is a natural number) inone turn by performing a complex Fourier transform in a circumferentialdirection on measured values of the wall thickness at a plurality ofpoints in a cross section of the pipe in an axial direction; secondcalculating means for calculating a thickness deviation amountindicating a degree of thickness deviation of each k-th-order thicknessdeviation, from an absolute value of the calculated complex Fouriercomponent; and third calculating means for calculating a position of athick portion or thin portion of each k-th-order thickness deviation,from a phase of the calculated complex Fourier component.
 30. Thethickness deviation information derivation apparatus as set forth inclaim 29, wherein the third calculating means calculates a positionargW(k) of the thick portion or a position argN(k) of the thin portionof each k-th-order thickness deviation with a unit of angle in which theposition of a first measurement point is 0°, using the real part R(k)and the imaginary part I(k) of the complex Fourier component of eachk-th-order thickness deviation, by${\arg\quad{W(k)}\frac{1}{k}\left( {\tan^{- 1}\frac{I(k)}{R(k)}} \right)\frac{180}{\pi}},{or}$${\arg\quad{N(k)}} = {\frac{1}{k}\left( {{\tan^{- 1}\frac{I(k)}{R(k)}} + \pi} \right){\frac{180}{\pi}.}}$31. The thickness deviation information derivation apparatus as setforth in claim 29, wherein the first calculating means calculates a realpart R(k) and an imaginary part I(k) of the complex Fourier component ofeach k-th-order thickness deviation by${{R(k)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\cos\left( {\frac{2\pi}{N}{k\left( {i - 1} \right)}} \right)}} \right\}}}},{and}$${I(k)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\sin\left( {\frac{2\pi}{N}{k\left( {i - 1} \right)}} \right)}} \right\}}}$where N is the number of measurement points at which the wall thicknessis measured in the cross section in the axial direction, and WT(i) is ameasured value of the wall thickness at the i-th measurement point amongthe measurement points aligned in the circumferential direction, and thesecond calculating means calculates a thickness deviation amount G(k) ofeach k-th-order thickness deviation byG(k)=4√{square root over (R(k)² +I(k)²)}.
 32. The thickness deviationinformation derivation apparatus as set forth in claim 31, wherein thethird calculating means calculates a position argW(k) of the thickportion or a position argN(k) of the thin portion of each k-th-orderthickness deviation with a unit of angle in which the position of afirst measurement point is 0°, using the real part R(k) and theimaginary part I(k) of the complex Fourier component of each k-th-orderthickness deviation, by${\arg\quad{W(k)}\frac{1}{k}\left( {\tan^{- 1}\frac{I(k)}{R(k)}} \right)\frac{180}{\pi}},{or}$${\arg\quad{N(k)}} = {\frac{1}{k}\left( {{\tan^{- 1}\frac{I(k)}{R(k)}} + \pi} \right){\frac{180}{\pi}.}}$33. A thickness deviation information derivation apparatus for derivinginformation about a thickness deviation that occurs in a pipe, based onmeasured values of wall thickness of the pipe, comprising: firstcalculating means for calculating a complex Fourier component of afirst-order thickness deviation of the wall thickness that changesperiodically once in one turn by performing a complex Fourier transformin a circumferential direction on measured values of the wall thicknessat a plurality of points in a plurality of cross sections of the pipe inan axial direction; second calculating means for calculating a thicknessdeviation amount indicating a degree of thickness deviation of thefirst-order thickness deviation, from an absolute value of thecalculated complex Fourier component, for the plurality of the crosssections in the axial direction; third calculating means for calculatinga phase of the first-order thickness deviation, from the calculatedcomplex Fourier component, for the plurality of the cross sections inthe axial direction; fourth calculating means for calculating a complexFourier component for each frequency representing the number of timesthe thickness deviation is twisted per unit length in a longitudinaldirection of the pipe, by performing a complex Fourier transform on acomplex function in which a complex number whose absolute value andphase are the thickness deviation amount and the phase thus calculated,respectively, is a function of position in the longitudinal direction ofthe pipe; and determining means for determining whether the values ofthe complex Fourier component in a plurality of predetermined frequencyranges are larger or not based on a predetermined boundary value. 34.The thickness deviation information derivation apparatus as set forth inclaim 33, wherein the first calculating means calculates a real partR(1) and an imaginary part 1(1) of the complex Fourier component of thefirst-order thickness deviation by${{R(1)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\cos\left( {\frac{2\pi}{N}\left( {i - 1} \right)} \right)}} \right\}}}},{and}$${I(1)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\sin\left( {\frac{2\pi}{N}1\left( {i - 1} \right)} \right)}} \right\}}}$where N is the number of measurement points at which the wall thicknessis measured in the cross section in the axial direction, and WT(i) is ameasured value of the wall thickness at the i-th measurement point amongthe measurement points aligned in the circumferential direction, thesecond calculating means calculates a thickness deviation amount r ofthe first-order thickness deviation byr=4√{square root over (R(1)² +I(1)²)}, the third calculating meanscalculates a phase θ of the first-order thickness deviation byθ=tan⁻¹ {I(1)/R(1)}, and the fourth calculating means performs a complexFourier transform on a function of y, f(y)=r(y)·exp(j·θ(y)), where j isan imaginary number, y is a length in the longitudinal direction of thepipe, and the r and θ are functions of y.
 35. A computer program productfor causing a computer to derive information about a thickness deviationthat occurs in a pipe, based on wall thickness values of the pipe,wherein said computer program product comprises a computer readablestorage medium having computer readable program code means embodied insaid medium, said computer readable program code means comprisingcomputer instruction means for: executing a first calculation step forcalculating a complex Fourier component of each k-th-order thicknessdeviation of the wall thickness that changes periodically k times (k isa natural number) by performing a complex Fourier transform on a seriesof the wall thickness values of the pipe; executing a second calculationstep for calculating a thickness deviation amount indicating a degree ofthickness deviation of each k-th-order thickness deviation, from anabsolute value of the calculated complex Fourier component; andexecuting a third calculation step for calculating a position of a thickportion or thin portion of each k-th-order thickness deviation, from aphase of the calculated complex Fourier component.
 36. The computerprogram product as set forth in claim 35, wherein the third calculationstep includes a step for calculating a position argW(k) of the thickportion or a position argN(k) of the thin portion of each k-th-orderthickness deviation with a unit of angle, using the real part R(k) andimaginary part I(k) of the complex Fourier component of each k-th-orderthickness deviation, by${\arg\quad{W(k)}\frac{1}{k}\left( {\tan^{- 1}\frac{I(k)}{R(k)}} \right)\frac{180}{\pi}},{or}$${\arg\quad{N(k)}} = {\frac{1}{k}\left( {{\tan^{- 1}\frac{I(k)}{R(k)}} + \pi} \right){\frac{180}{\pi}.}}$37. The computer program product as set forth in claim 35, wherein thefirst calculation step includes a step for calculating a real part R(k)and an imaginary part I(k) of the complex Fourier component of eachk-th-order thickness deviation by${{R(k)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\cos\left( {\frac{2\pi}{N}{k\left( {i - 1} \right)}} \right)}} \right\}}}},{and}$${I(k)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\sin\left( {\frac{2\pi}{N}{k\left( {i - 1} \right)}} \right)}} \right\}}}$where N is the number of the wall thickness values, and WT(i) is thei-th wall thickness value, and the second calculation step includes astep for calculating a thickness deviation amount G(k) of eachk-th-order thickness deviation byG(k)=4√{square root over (R(k)² +I(k)²)}.
 38. The computer programproduct as set forth in claim 37, wherein the third calculation stepincludes a step for calculating a position argW(k) of the thick portionor a position argN(k) of the thin portion of each k-th-order thicknessdeviation with a unit of angle, using the real part R(k) and imaginarypart I(k) of the complex Fourier component of each k-th-order thicknessdeviation, by${\arg\quad{W(k)}\frac{1}{k}\left( {\tan^{- 1}\frac{I(k)}{R(k)}} \right)\frac{180}{\pi}},{or}$${\arg\quad{N(k)}} = {\frac{1}{k}\left( {{\tan^{- 1}\frac{I(k)}{R(k)}} + \pi} \right){\frac{180}{\pi}.}}$39. A computer program product for causing a computer to deriveinformation about a thickness deviation that occurs in a pipe, based onwall thickness values of the pipe, wherein said computer program productcomprises a computer readable storage medium having computer readableprogram code means embodied in said medium, said computer readableprogram code means comprising computer instruction means for: executinga first calculation step for calculating a complex Fourier component ofa first-order thickness deviation of the wall thickness that changesperiodically once by performing a complex Fourier transform on each of aplurality of series of the wall thickness values; executing a secondcalculation step for calculating a thickness deviation amount indicatinga degree of thickness deviation of the first-order thickness deviation,from an absolute value of the calculated complex Fourier component, foreach of a plurality of series of the wall thickness values; executing athird calculation step for calculating a phase of the first-orderthickness deviation, from the calculated complex Fourier component, foreach of a plurality of series of the wall thickness values; executing afourth calculation step for calculating a complex Fourier component foreach frequency by performing a complex Fourier transform on arelationship between the thickness deviation amount and the phasecalculated for each of a plurality of series of the wall thicknessvalues, over a plurality of series of the wall thickness values; andexecuting a determination step for determining whether values of thecomplex Fourier component in a plurality of predetermined frequencyranges are larger or not based on a predetermined boundary value. 40.The computer program product as set forth in claim 39, wherein the firstcalculation step includes a step for calculating a real part R(1) and animaginary part I(1) of the complex Fourier component of the first-orderthickness deviation by${{R(1)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\cos\left( {\frac{2\pi}{N}\left( {i - 1} \right)} \right)}} \right\}}}},{and}$${I(1)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left\{ {{{WT}(i)}{\sin\left( {\frac{2\pi}{N}1\left( {i - 1} \right)} \right)}} \right\}}}$where N is the number of the wall thickness values included in a seriesof the wall thickness values, and WT(i) is the i-th wall thickness valueincluded in the series of the wall thickness values, the secondcalculation step includes a step for calculating a thickness deviationamount r of the first-order thickness deviation byr=4√{square root over (R(1)² +I(1)²)}, the third calculation stepincludes a step for calculating a phase θ of the first-order thicknessdeviation byθ=tan⁻¹ {I(1)/R(1)}, and the fourth calculation step includes a step forperforming complex Fourier transform on a function of y,f(y)=r(y)·exp(j·θ(y)), where j is an imaginary number, y is a length ina longitudinal direction of the pipe, and the r and θ are functions ofy.